Statistical properties of photons in a three-level plus two-mode model

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Statistical properties of photons in a three-level plus two-mode model

N.N. Bogolubov, Fam Le Kien, A.S. Shumovsky

To cite this version:

N.N. Bogolubov, Fam Le Kien, A.S. Shumovsky. Statistical properties of photons in a three-level plus two-mode model. Journal de Physique, 1986, 47 (3), pp.427-435. �10.1051/jphys:01986004703042700�.

�jpa-00210222�

Statistical properties ^of photons ^{in a} three-level plus ^{two-mode model}

N. N. Bogolubov Jr., ^{Fam Le} Kien and A. S. Shumovsky

Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,

P.O. Box 79, 101 000, Moscow, ^U.S.S.R.

(Reçu ^le 22 juillet ^1985, révisé le 12 novembre, accepté le 19 novembre 1985)

Resume.

On étudie les caractéristiques ^des propriétés statistiques ^des photons ^{dans un modèle} analytique ^à

trois niveaux et deux modes du champ. Le modèle consiste en un émetteur unique, à trois niveaux en lambda, initialement dans le niveau le moins excité, interagissant ^avec deux modes du champ quantifié. ^{On calcule les} variances, ^{en ordre} normal, ^{du nombre de} photons ^et les fonctions de corrélation croisées. Tous les résultats sont obtenus sans utiliser, ^{ni théorie de} perturbation, ⁿⁱ d’approximations ^de décorrélation. On montre que les états du mode « signal » ^du champ, ^{obtenus à} partir ^{du vide} initial, ^sont sub-poissoniens ^{à tout instant On} décrit l’évo- lution des propriétés statistiques ^des photons quand ^{le mode «} signal » ^{et le mode «} pompe » ^sont initialement dans un état cohérent de Glauber. Dans des conditions convenables, ^on pourrait obtenir dans ce système, ^un dégroupement ^de photons ^{et une} anti-corrélation entre les modes.

Abstract

The characteristics of the photon statistical properties ^{are studied in a soluble} three-level plus ^two-

mode model. The model consists of a single lambda-configuration three-level emitter interacting ^{with two modes}

of the quantized ^radiation field, ^{the emitter} being initially in the lowest level. The normally ordered variances of the photon numbers and the cross-correlation function are calculated All the results are obtained without using perturbation theory ^and decorrelation approximations. It is shown that the states of the signal-mode field pro- duced from its initial vacuum are sub-Poissonian for all times. For the case when both the signal-mode and pump- mode fields are initially in the Glauber coherent states the temporal ^{behaviour of the} photon statistical characteris- tics are described Under suitable conditions photon antibunching and anticorrelation between the modes are

potentially possible ^{in the} system.

Classification

Physics ^Abstracts

32.80

42.50

1. Introduction.

During ^{ten last} years the photon antibunching ^effect

has attracted considerable interest, since it has no

counterpart in classical electrodynamics and, hence, provides ^new evidence of the wave-particle ^dualism.

Photon antibunching is characterized by ^{a nonclassi-}

cal state of the field in which the variance of the num-

ber of photons is less than the mean number of pho-

tons, ^{i.e. the} photons exhibit sub-Poissonian statistics.

While experiments ^{thus far are} restricted to the study

of resonance fluorescence from single ^atoms [1-3],

various physical mechanism that generate ^{such a}

state have been shown to be theoretically possible (see ^{Refs. [4-20]} and the reviews [21, 22]). Among

them are second harmonic generation [4-7] ^{and three-}

wave parametric interaction [8] ^{in a nonlinear} medium,

resonance fluorescence from two-level atoms [9-15], degenerate ^four-wave mixing [16], two-photon absorp-

tion [17, 18], free-electron-laser generation [19, 20].

Recently, ^the generation ^of squeezed states, another class of nonclassical states [28], has also become the

object of active researches (see ^{[28-30] and refs.}

therein).

In this _paper the production of sub-Poissonian non-classical states by using ^two-mode processes in a three-level system ^{will be} discussed. On the basis of the exactly ^soluble model, studied in [23] ^{we shall}

examine the time behaviour of the normally ^ordered

variances of the photon numbers and of the ^cross correlation between the modes. It will be shown that under suitable conditions photon antibunching ^and

anticorrelation between the modes are possible ⁱⁿ

the system.

The remainder of the _paper is organized ^{as follows.}

In section 2 the model Hamiltonian and the _necessary

previous ^{results are} given. ^{In section 3 we} study ^the

time behaviour of the normally ordered variances of the photon ^{numbers. In section 4 the cross} correlation between the modes is calculated Finally, the conclu- sions are briefly summarized in section 5.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01986004703042700

428

2. Model Hamiltonian and basic equations.

^The lambda-configuration three-level emitter considered here is shown in figure ^{1. The} upper level 3 is connected with the levels 1 and 2 by dipole transitions whereas the transition 1-2 is forbidden. The Hamiltonian

describing ^the interaction of the emitter with the two-mode resonant radiation field in the dipole ^and rotating ^wave approximations ^is [23]

Here, ^the operator Rjj = I j) ^{j I} describes the

population ^of level j ^{with the} corresponding energy

1iQj and state vector I j). Rij * [ i ) j I ^{is the} ope- rator of transition from level j ^{to level i. The} operators

Rij, i, j ⁼ 1, 2, 3, obey ^the following relations :

The photon operators âa, â: ^{describe two modes of}

the radiation field with the ^resonance frequencies (J)a = D3 - Da, ^and ga’s ^{are the} corresponding para- meters of emitter-mode coupling.

The three-level two-mode model (1) ^has extensively

been investigated [31, 23] (see ^also [32] ^{and Refs.}

therein). ^{In the} framework of this model one can

study ^not only ^various purely quantum effects in

population dynamics ^and photon statistics [31, 34]

but also fine features in indirect two-mode interaction

Fig. ^1.

Energy ^{level structure} of the lambda configuration

emitter interacting ^{with the two-mode resonant radiation}

field.

via the emitter. An excellent review of the dynamical theory of the model has recently ^been given by ^Yoo

and Eberly ⁱⁿ paper [31]. Nonclassical photon ^sta-

tistics has been numerically ^shown. However, ^these authors do not make _any comments about it. Some discussions about experimental verifications, being possible ^{due to} progress in the realization of a single

atom in a cavity [33-37], have been given ^{in refe-}

rence [31]. ^{We shall} study ^{the time} behaviour of

photon statistics and mode correlations below.

We assume that the emitter is initially ^{on the}

lowest level 1. In the _papers [23] ^the time-dependent

two-mode photon-distribution ^function Plnl’ n2) ^has

been found explicitly ^and rigorously. ^{It reads}

where

and the function P(ni, n2) ⁼ Pt=O(nl’ n2) is the initial photon distribution. In general case, for an arbitrary

initial field state described by ^a density ^matrix PF ^{we have}

Starting ^from equations (3. 5), ^{we shall examine} photon statistics of the modes in the remainder of this

paper.

3. Normally ordered variances of the numbers of the photons ^{in the modes.}

From equation (3) ^one easily ^{finds the} normally ordered variance Vjsj ^{of the photon} number in the signal ^mode (mode 2)

The expression ^{of the} corresponding quantity V,Pl ^{for the pump mode (mode 1)} is found from equation (3) ^{to be}

Here,

and

denote the mean numbers of the initial photons ⁱⁿ

the _pump and signal modes, respectively.

As is well known, ^the quantities Vjsj ^and V [P)

characterize the photon statistical properties ^{of the}

modes. They ^are proportional ^{to the excess coinci-}

dence counting ^{rates to} be measured in a Hanbury

Brown and Twiss-type [24, 25] experiment. ^The sign plus ^{or minus or} Vjsj ^(or V[P]) ^{shows the photon sta-}

tistics of the signal (or pump) ^{mode is} super- ^or sub- Poissonian and indicates, therefore, ^whether bunching (+) ^or antibunching (-) ^occurs [21, 22]. Now, ^we consider some consequences of the expressions (6)

and (7).

First of all note that if the signal ^{mode 2 is} initially

in the vacuum state, i.e. P(nl, n2)

bn2o ^{P(ni), from}

(6) ^{we have}

This implies ^{that the} signal light ^field generated ^from

the initial vacuum by arbitrary light pumping ^has

sub-Poissonian photon statistics and, hence, ^exhibits photon antibunching for all times t > 0. In figure ²

we plot V[S] ^{versus time go t for the case} of coherent

pumping ^with

Fig. ^2.

Temporal behaviour of the normally ^ordered

variance V[S](t) ^{of the photon} number in the signal-mode

fied produced ^{from the vacuum} by Glauber-coherent

pumping. Calculations are performed for g1

g2 ⁼ go,

P(n1’ n2) = ðn20 P(n1) P(n1) = exp(- ’l) fill’ll ^and n1

12. Time is measured in units of 1/go.

and n-1

^{12. The} figure ^shows distinctly ^{the Rabi}

non linear oscillations, collapse and revival of their

envelopes (see [26] ^and below).

Let us now assume that both the signal ^and pump modes are initially in Glauber states I z, > ^and z2 >, respectively. ^{This means,} the initial distribution

P(ni, n2) ^{of the} photon numbers in the modes is given

430

by

Substituting (9) ^into (6) ^and (7), ^we perform ^{a nume-}

rical examination for visi ^and vipi ^{In figure 3 we}

plot ^these quantities ^as functions of time for g2 ⁼

29t ⁼ 2 go, n2 = 9, ii, ^{= 4. For} very short times

(0 go t 0.25), the value of V[s] is ^{positive, whilst}

the value of V [P] ^{is negative. As time goes on (go t 3),}

we observe oscillations of both the values and their

corresponding signs (see Fig. 3a). ^{From the} figure ^{it is}

clear that sub-Poissonian photon statistics and anti-

bunching ^{in the modes occur for some} finite inter- vals. Such a time behaviour of photon statistics is connected with the so-called Rabi oscillations [21, 22, 26]. ^For longer ^times (go t > 3) Vjsj and V [P]

reach the steady ^{values. This} corresponds ^{to the} Cummings collapse region ^{26. The} negative steady

value Vo predicts antibunching ^{in the} signal ^mode.

The stationary behaviour of the variances continues until the appearance of revivals [26], ^see figure 3b,

where they ^start oscillating again. Thus, ^{we see that}

in the considered case, when g2 n2/gî ni ^{= 9 » 1} antibunching ^{in the} signal mode takes place ^both

in the Rabi oscillation region ^and during ^the collapse

and revival. An analytic approximate description ^of

the above-obtained numerical results can be given

in the following.

Due to the well-known properties ^{of the two-}

dimension Poissonian distribution (9)

from equations (6) ^and (7) ^{we have}

Here, for convenience the notation (...) == L P(nl, n2) (...) ^{has been introduced}

Ill,n2

In the case nl’ n2 > ¹ the summations over n, and n2 in (11) ^{can be} performed approximately by using saddle-point techniques [26] ^{if one} notices that the weighting ^factor P(ni, n2) ^will peak ^at (n1, n2) ^with relatively

narrow dispersion. ^{This means that the} expressions ⁱⁿ (11) ^{can be} changed ^to integrals ^{of fast} oscillating ^functions

which after trivial transformations can be written as

with (see [26])

Note that the function F(t; g2) approaches unity ^{in the limit t -+ oo. Then we can} easily ^{find from} equations (12)

analytical approximate expressions ^{for the} steady ^values V0[s] ^and VI ^{of the} variances. They ^read

V,, ^-

Fig. ^3.

^Time dependence ^{of the} normally ^{ordered va-}

riances V.(i) ^and _Vlp,(t) ^{of the} photon numbers in the signal

and pump modes, respectively. Both the fields of the modes

are initially in the Glauber coherent states with nl

4,

n2

9, gt

0.5 g2 ^z go. Time is measured in units of

1/go. (a) Short times : go t ^{5. In this case the basic} oscillations occur at the Rabi frequencies, ^{and are} subject

to Gaussian collapse resulting ^{in the} steady ^behaviour.

(b) Long times : the range of time is eight ^times larger ^{than in} (a). ^The long time behaviour is characterized by ^revival.

We plot ^the quantity Vo ^{as a function of n2 in figure 4a}

(for ni ⁼ 10, gt ⁼ 92 = go) ^{and as a} function of

n1 ⁱⁿ figure ^4b (for n2 ^{= 0,10,} gi = g2 ⁼ go). ^From

Fig. ^4.

Steady ^value V&J ^{of the normally} ordered variance of the photon number in the signal ^mode. (a) ^The depen-

dence of V&J upon n2 in the case 91

g2, n1

^10. (b) ^The dependence ^of Vo ^upon nl ^{in the cases} when g1 = 92’

n2 = 0 ^{and 10.}

the figures ^{we see that} Vo ^{is negative for very small}

values n2 (n2 ⁼ 0) ^{and for} large ^values n2 ^as compar- ed with ii, 2 n2 g21 n1). By using ^{the first} equa- tion in (15) ^{we can} easily get the estimate

for the case g2 - >> g2 -1. ^{On the other hand, for}

432

the case when the initial state of the signal ^{mode field}

is vacuum or near-vacuum, Le. n2 ^{0, and when} n, > ^{1, we find} straight forwardly ^from equation (11)

the evaluation

(which is also in compliance with the first equation

in (15) in the limit n-2 --+ 0). ^The inequalities ⁱⁿ equa- tions (16) ^and (17) ^{mean that} photon statistics of the

signal mode field in the steady regime will be sub- Poissonian if either g’ 2 n- 2 >> 9 2 1 n, (n-,, n- 2 1) ^or n2 - 0, Ei » ^{1. This confirms} the numerical results obtained above, ^see figures 2, ^{3 and 4.}

For short times and large ^initial photon ^numbers ni, n2 by using ^the saddle-point method for evaluating

the integral (13), ^{one can} get

The right-hand ^{side of} equation (18) behaves like

damped ^cosine oscillations which are terminated

by the Gaussian envelope exp[ - t2/(2 t2c)]. ^Here

the collapse ^time Tc is defined as

The approximate expression (18) together ^with equa- tions (12) give ^a description of the behaviour of the variances V[s], V [P] in the short time regime t l’ c

(see Fig. 3a). ^In particular, ^for early ^{time t = 0 we}

can obtain

Emphasize ^{that the} approximate expression (18)

is not valid in the long ^time region. Therefore, ^it

cannot describe the revival of the oscillation envelopes

of VIS] ^and V[P], ^{which occurs when t} >> LC’ ^see figure ^3b.

Applying ^the long-time approximate method deve-

loped ^on the basis of the saddle-point ^method by Eberly ^and his coworkers in reference [26] ^{to calcu-}

late the integral (13) ^{we can} get ^{in the} special case 91 = 92 =- go the following ^evaluation

where the notation

has been introduced, ^and

The approximate expression ^{(21) is rather} good for all times [26]. ^It indicates the periodicity of the revivals. The interval TR between the revivals of the oscillation envelope of F(t; g2) ^can be found from the second equation

in (22) ^{to be}

4. Cross correlation.

More interesting ^{is the cross} correlation between the two modes. Its magnitude ^can be characterized by

The quantity Vcross is proportional ^{to the excess} coincidence counting ^{rate to} be measured in an experiment

of the Hanbury Brown-Twiss-type ^{with two different} light ^beams [22, 27]. ^We speak ^of anticorrelation, ^when Vcross ^becomes negative. Anticorrelation has been observed in light scattering ^from nonspherical particles ⁱⁿ

dilute solution [27]. ^{We consider now the time behaviour of} Vcross in the model (1).

Utilizing equation (3) ^{and the} definition (25) ^{we can} get

Equation (26) together ^with equations (4) ^{allow us to} describe the time evolution of the cross correlation in the

general ^case of the initial field state (5). ^{Let us assume} that the initial photon distribution P(n1, n2) ^is given by (9).

This means that the signal ^and pump fields are coherent and statistically independent ^{at the} beginning ^{of the}

interaction. Then, ^{due to the} properties (10), ^{we find from} (26) ^{the result}

Here (...) indicates, ^as in section 3, averaging ^{over the} distribution (9).

A numerical computation of the formula (27) ^{has been} performed ^{for the case, for} example, when g1

2g2 =

2 go, ni ^{= 12,} n2 ⁼ 3. The results are plotted ⁱⁿ figure ^{5. We see from the} figure ^that anticorrelation occurs for

some finite intervals of times in the Rabi oscillation region, ^and during ^the steady regime. The successive _pre-

sences of correlation and anticorrelation in the Rabi oscillation region ^{are noted For} large photon ^numbers ni, n2 by using ^the approximate ^formulae

obtained from (4), ^{we can} apply ^{the same} analytical description ^we have used in the previous ^{section to} (27).

The collapse ^and revival of Vcross ^are easily ^described by utilizing ^the approximate expressions (18) ^and (21).

Due to that the function F(t; g2) approaches unity ^{in the} collapse region, ^see (18), and in the region ^of infinitely large times, ^see (21), ^the steady ^{value of} V cross is quickly ^{found to be}

Fig. ^5.

Time evolution of the cross-correlation function

Vcross(t). Both the fields of the modes are initially ^{in the}

Glauber coherent states with nl

^12, n2

3, gl

2 g2 ⁼

2 go. Time is measured in units of 1/g0.

434

Hence, ^{for the case} g’ 1 ;il >> g2 n2 ^one gets

The inequality ⁱⁿ equation (30) indicates the anti- correlation between the signal and pump modes in the steady regime. ^This analytical prediction ^{is in} compliance ^{with the numerical result obtained above,}

see figure ⁵ (where gf nl/gi n2 ⁼ 16). ^{In the} opposite

limit case g2 jil g2 n2 ^{one has}

that indicates the correlation between the modes.

Finally, ^for very short times t 0 we find from

equations (27) ^and (4) the formula

which describes the appearance of the correlation between the modes due to the interaction with the

emitter, ^see figure ^5.

5. Conclusions.

In this _paper we have studied the characteristics of the photon statistical properties ^{in a soluble three-}

level plus two-mode model. The normally ^ordered

variances of the photon numbers and the ^cross- correlation function have been calculated It has been

shown that the states of the signal mode field _pro- duced from its initial vacuum are sub-Poissonian for all times. This production ^of sub-Poissonian states

does not depend ^upon the initial state of the _pump mode field

The quantum collapse and revival behaviour of the

photon statistical characteristics in the lossless model considered here have been noted

For the case when both the signal-mode ^and pump- mode fields are initially ^{in the} Glauber coherent

states the time behaviour of the photon-number

variances and of the ^cross correlation between the modes has been numerically ^{examined and} analyti- cally approximated The theoretical investigations

show that under suitable conditions the occurrence

of photon antibunching and the presence of anti- correlation between the modes are potentially possible

in the system. ^{It has been} established that

(i) ^{In the} region of short times (the ^{Rabi oscilla-}

tion region) photon statistics of the signal-mode

field is sub-Poissonian for some finite intervals;

(ii) Photon statistics of the signal-mode field in the

steady regime ^{will de} sub-Poissonian if either g’ - 91 2 ;il (n-1, ;i2 > 1) or ;i2 = 0 (n-, >> 1);

(iii) ^{Due to} the interaction with the emitter the correlations between the modes ^come into existence.

The anticorrelation between the modes will occur in the steady regime if g 1 n 1 > g2 n2 (n-" - n2 >> 1 ).

All the results have been obtained without using perturbation theory ^and decorrelation approxima-

tions.

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